MinT - Best Known (121−23, 121, s)-Nets in Base 4

Best Known (121−23, 121, s)-Nets in Base 4

(121−23, 121, 1490)-Net over F₄ — Constructive and digital

Digital (98, 121, 1490)-net over F₄, using

4¹ times duplication [i] based on digital (97, 120, 1490)-net over F₄, using
- net defined by OOA [i] based on linear OOA(4¹²⁰, 1490, F₄, 23, 23) (dual of [(1490, 23), 34150, 24]-NRT-code), using
  - OOA 11-folding and stacking with additional row [i] based on linear OA(4¹²⁰, 16391, F₄, 23) (dual of [16391, 16271, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
      1. linear OA(4¹²⁰, 16384, F₄, 23) (dual of [16384, 16264, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      2. linear OA(4¹¹³, 16384, F₄, 22) (dual of [16384, 16271, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      3. linear OA(4⁰, 7, F₄, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(121−23, 121, 8199)-Net over F₄ — Digital

Digital (98, 121, 8199)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹²¹, 8199, F₄, 2, 23) (dual of [(8199, 2), 16277, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4¹²¹, 16398, F₄, 23) (dual of [16398, 16277, 24]-code), using
  - discarding factors / shortening the dual code based on linear OA(4¹²¹, 16399, F₄, 23) (dual of [16399, 16278, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(20) [i] based on
      1. linear OA(4¹²⁰, 16384, F₄, 23) (dual of [16384, 16264, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      2. linear OA(4¹⁰⁶, 16384, F₄, 21) (dual of [16384, 16278, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      3. linear OA(4¹, 15, F₄, 1) (dual of [15, 14, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(121−23, 121, 6050892)-Net in Base 4 — Upper bound on s

There is no (98, 121, 6050893)-net in base 4, because

1 times m-reduction [i] would yield (98, 120, 6050893)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 766847 766118 205756 494978 287875 842152 123461 182929 589989 957115 805308 571580 > 4¹²⁰ [i]